Can't snap two objects without space between
Are you snapping in Object Mode?  If so, the only snap source will be the pivot point of the object you are moving. 

If you snap in Edit (Vertex) Mode, switch the snap settings to target 'Vertices', with the source set to 'Active'. Select all the vertices in the moving object, and Shift right-click the vertex you would like to be the snap source twice to make it active. Now with snap switched on, move the active vertex over the vertex you would like it to match.. it will bring the other vertices along with it.